The Compleat Distiller

THE COMPLEAT DISTILLER 102
Since a quantity of liquid L with mol fraction X of a component
contains LX mols of that component, the balance equation at
plate N for the volatile component we're considering is:
Total mols received = Total mols lost
V N-1 Y N-1 + L N+1 X N+1 = V N Y N + L N X N
This equation will apply at every point in the column if it's in
balance (ie. in equilibrium). You just have to change the suffixes
to match the plate number.
The condenser on top of the compound column is considered the
top plate, and the boiler is the bottom plate. The liquid returned
to the column from the top condenser is a special case of
'imposed' reflux, unlike the 'natural' reflux generated inside the
column. We've therefore set it apart by identifying the final mol
fraction of the liquid coming from the condenser as X D . So, if
X D is the final mol fraction of the liquid condensed at the top of
the column, and L N+1 is the reflux, then
X N+1 = X D
The balance equation for plate N may be written:
V N-1 Y N-1 + L N+1 X D = V N Y N + L N X N
You can consider the whole column and as one block unit.
Material entering this block = V 0 Y 0 +L N+1 X N+1
Material leaving the block is = V N Y N + L 1 X 1
Fig. 8-9
Which means that
V 0 Y 0 +L N+1 X D = V N Y N + L N X N
But V N Y N = L N+1 + DX D (consider what enters and leaves the condenser).
So
V 0 Y 0 = L 1 X 1 + DX D
This describes the balance of the column as a whole.